analysis of combined conductive and radiative heat
TRANSCRIPT
ANALYSIS OF COMBINED CONDUCTIVE AND RADIATIVE HEAT TRANSFER IN A TWO DIMENSIONAL SQUARE ENCLOSURE USING THE FINITE VOLUME METHOD
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology in
Mechanical Engineering
By
Prabir Kumar Jena
Department of Mechanical Engineering National Institute of Technology
Rourkela 2009
ANALYSIS OF COMBINED CONDUCTIVE AND RADIATIVE HEAT TRANSFER IN A TWO DIMENSIONAL SQUARE ENCLOSURE USING THE FINITE VOLUME METHOD
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology in
Mechanical Engineering
By
Prabir Kumar Jena
Under the guidance of Prof. Swarup Kumar Mahapatra
Department of Mechanical Engineering National Institute of Technology
Rourkela 2009
National Institute of Technology Rourkela
CERTIFICATE
This is to certify that the thesis entitled “ANALYSIS OF COMBINED
CONDUCTIVE AND RADIATIVE HEAT TRANSFER IN A TWO DIMENSIONAL
SQUARE ENCLOSURE USING THE FINITE VOLUME METHOD” submitted by Mr.
Prabir Kumar Jena in partial fulfillment of the requirements for the award of Master
of Technology Degree in Mechanical Engineering with specialization in Thermal
Engineering at the National Institute of Technology, Rourkela (Deemed University)
is an authentic work carried out by him under my supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been
submitted to any other University/ Institute for the award of any degree or diploma.
Date: Prof. SWARUP KUMAR MAHAPATRA
Department of Mechanical Engineering National Institute of Technology Rourkela – 769008
ii
CONTENTS CERTIFICATE i
CONTENTS ii
ACKNOWLEDGEMENT iv
ABSTRACT v
LIST OF FIGURES vi
LIST OF TABLES viii
NOMENCLATURE ix
CHAPTERS
1. INTRODUCTION AND LITERATURE REVIEW 1
2. DISCRETIZATION METHODS BY USING FVM
2.1 The nature of numerical methods 6
2.1.1 The discretization concept 6
2.1.2 The structure of the discretization equation 7
2.2 Methods of deriving the discretization equations 7
2.3 Control-volume formulation 8
2.3.1 Finite volume: basic methodology 9
2.3.2 Cells and nodes 9
2.4 Finite volume method for unsteady flows 11
2.4.1 One-dimensional unsteady heat conduction 11
2.4.2 Explicit scheme 13
2.4.3 The fully implicit scheme 14
2.4.4 Crank-Nicholson scheme 15
2.5 Two-dimensional unsteady heat conduction 16
iii
2.6 Solutions of algebraic equations 20
2.6.1 The Tri-diagonal matrix algorithm 21
2.7 Computational domain 23
2.8 Flow chart for solving transient conduction 23
3. RADIATIVE HEAT TRANSFER
3.1 Mathematical description of radiation 25
3.2 Radiative properties of materials 26
3.2.1 Black body radiation 26
3.2.2 Surface properties 27
3.3 Formulation of RTE by using FV method 28
3.4 Solution Procedure 34
4. ANALYSIS
4.1 Energy equation 36
4.2 Calculation of incident radiation and heat flux 36
5. RESULTS AND DISCUSSION 39
6. CONCLUSIONS AND FUTURE SCOPE
6.1 CONCLUSIONS 57
6.2 FUTURE SCOPE 57
7. LIST OF REFERENCES 59
iv
ACKNOWLEDGEMENT
I am extremely fortunate to be involved in an exciting and challenging research project like
“analysis of combined conductive and radiative heat transfer in a two dimensional square
enclosure using the finite volume method”. This project increased my thinking and
understanding capability as I started the project from scratch.
I would like to express my greatest gratitude and respect to my supervisor Prof. Swarup
Kumar Mahapatra, for his excellent guidance, valuable suggestions and endless support. He
has not only been a wonderful supervisor but also a genuine person. I consider myself
extremely lucky to be able to work under guidance of such a dynamic personality. Actually
he is one of such genuine person for whom my words will not be enough to express.
I would like to express my sincere thanks to Prof. R.K.Sahoo for his precious suggestions and
encouragement to perform the project work. He was very patient to hear my problems that I
am facing during the project work and finding the solutions. I am very much thankful to him
for giving his valuable time for me.
I would like to express my thanks to all my classmates, all staffs and faculty members of
mechanical engineering department for making my stay in N.I.T. Rourkela a pleasant and
memorable experience and also giving me absolute working environment where I unlashed
my potential .
I want to convey my heartiest gratitude to my parents for their unfathomable encouragement.
Date: Prabir Kumar Jena Roll. No. 207ME310 M.Tech. (Thermal)
v
Abstract
An efficient tool to deal with multidimensional radiative heat transfer is in strong demand to
analyse the various thermal problems combined either with other modes of heat transfer or with
combustion phenomena. Combined conduction and radiation heat transfer without heat
generation is investigated. Analysis is carried out for both steady and unsteady situations.
Two-dimensional gray Cartesian enclosure with an absorbing, emitting, and isotropically
scattering medium is considered. Enclosure boundaries are assumed at specified temperature.
The finite volume method is used to solve the energy equation and the finite volume method
is used to compute the radiative information required in the solution of energy equation. The
Implicit scheme is used to solve the transient energy equation. Transient and steady state
temperature and heat flux distributions are found for various radiative parameters.
In the last two decade, finite volume method (FVM) emerged as one of the most
attractive method for modeling steady as well as transient state radiative transfer. The finite
volume method is a method for representing and evaluating partial differential equations as
algebraic equations. Similar to the finite difference method, values are calculated at discrete
places on a meshed geometry. "Finite volume" refers to the small volume surrounding each
node point on a mesh. In the finite volume method, volume integrals in a partial differential
equation that contain a divergence term are converted to surface integral using divergence
theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume.
Because the flux entering a given volume is identical to that leaving the adjacent volume,
these methods are conservative.
In the present work the boundaries are assumed to be gray and the medium scatters
isotropically. The current study examines the finite volume method (FVM) for coupled radiative
and conductive heat transfer in square enclosures in which either a non-scattering or scattering
medium is included. Implicit Scheme has been used for solving the coupled conductive and
radiative heat transfer equation. The transient temperature distributions are studied for the
effects of various parameters like the extinction coefficient, the scattering albedo and the
conduction radiation parameter for both constant. The effect of radiative parameters on the
system to reach the steady state is also studied.
vi
LIST OF FIGURES Figure No. Title Page No.
2.1 Three successive grid points used for the Taylor-series expansion 8
2.2 Cells and Nodes of Control Volume 10
2.3 Typical Control Volume 10
2.4 One dimensional control volume 11
2.5 Two Dimensional Control Volume 17
2.6 Computational Domain 23
3.1 Typical control angle 30
3.2 Intensity Il in direction lΔΩ in the center of the elemental
Sub-solid angle Ωl 30
3.3 Control angle orientations 32
3.4 Flowchart for overall solution procedure 34
5.1 Temperature isotherms at steady state 39
5.2 Variation of Heat flux at the bottom wall along X-axis 40
5.3 Variation of heat flux at the top cold wall along X-axis 40
5.4 Variation of heat flux at the side cold wall along Y-axis 41
5.5 Dimensionless temperature profiles for N=1 42
5.6 Dimensionless temperature profiles for N=0.1 42
5.7 Dimensionless temperature profiles for N=0.01 43
5.8 Dimensionless temperature profiles for N=0.001 43
5.9 Variation of fractional radiative heat flux for N=1 44 5.10 Variation of fractional radiative heat flux for N=0.1 45 5.11 Variation of fractional radiative heat flux for N=0.01 46 5.12 Variation of fractional radiative heat flux for N=0.001 47
5.13 Variation of the total heat flux for wall emissivity 48
of εw=1 at the bottom
5.14 Variation of total heat flux for εw=0.8, at the bottom wall, Y=0. 48
5.15 Variation of total heat flux for εw=0.5, at the bottom wall, Y=0 49
5.16 Variation of total heat flux for εw=0.2, at the bottom wall, Y=0. 49
5.17 Temperature isotherms for N=0.001 and ω=0. 50
vii
LIST OF FIGURES
Figure No. Title Page No.
5.18 Temperature isotherms for N=0.001 and ω=0.5. 51
5.19 Temperature isotherms for N=0.001 and ω=1. 51
5.20 Temperature isotherms for N=0.001 and ω=0 and ε=0.5. 52
5.21 Temperature isotherms for N=0.001 and ω=0.5 and ε=0.5. 52
5.22 Number of time steps = 20. 53
5.23 Number of time steps=40. 53
5.24 Number of time steps=80. 54
5.25 Temperature Variation with time 54
viii
LIST OF TABLES
Table No. Title Page No.
5.1 Variation in time period required to reach at steady state. 55
ix
Nomenclature
K Thermal conductivity
N Conduction-Radiation parameter
ρ Density of the medium
qr Radiative heat flux
qc Conductive heat flux
a coefficient of discretization equation
b source term in discretization equation
c the speed of light
cycx DD ′′ , direction cosine in x , y direction respectively
G incident radiation
I actual intensity
M total number of control angels
n unit outward normal vector of the control volume face
q heat flux
s distance
S source function
lmS modified source function
s unit direction vector
T temperature
t time
ζ Non-dimensional time
x, y, z coordinate direction
β extinction coefficient
lmβ modified extinction coefficient
AΔ area of control volume faces
x
vΔ volume of control volume
yx ΔΔ , x and y direction control volume width
ΔΩ control angle
ε emissivity
θ polar angle measured from ze
κ absorption coefficient
sσ scattering coefficient
σ Stefan-Boltzmann’s coefficient
Φ scattering phase function
ll′Φ average scattering phase function
φ azimuthal angle measured from xe
Subscripts
b black body
D downstream
m modified
P node
U upstream
E, W, N, S east, west, north, south neighbors of P
e, w, n, s east, west, north, south direction
superscripts
ll ′, angular directions
0 value from previous iteration
* non-dimensional quantities
CHAPTER 1
INTRODUCTION AND
LITERATURE REVIEW
2
INTRODUCTION AND LITERATURE REVIEW
Efficiency improvements in most industrial thermal processes are achieved by
increased process temperatures. The optimum design of such processes is leading to
temperature loads near the existing material limits. In recent years, high-temperature-resistant
materials, made not only of ceramics but also of special alloys, have become available to
realize high-temperature furnaces, gas turbine combustion chambers and blades, heat
exchangers, porous volumetric solar receivers, surface and porous IR-radiant burners.
Experimental and numerical investigations are necessary in order to perform an optimum
design and to elucidate the advantages of the temperature increase of such improved
processes utilizing advanced high-temperature materials. Numerical computations of flow
with heat and mass transfer in such high-temperature appliances require a through
consideration of radiative heat transfer.
Radiation either combined with other modes of heat transfer or with combustion
phenomena in a multidimensional enclosure such as a combustion chamber, furnace and
porous medium has received much attention due to a realization of its importance in the
associated application fields. However, since an exact analytical solution to the highly non-
linear integro differential radiative transfer equation (RTE) is nearly impossible to find, an
efficient tool to deal with multidimensional radiative heat transfer is in strong demand to
analyse various thermal problems.
Analysis of combined conduction and radiation heat transfer in a radiatively
participating medium has numerous engineering applications. Examples of such applications
are in the analysis of heat transfer through high temperature energy conversion devices,
semitransparent materials, fibrous and foam insulations, porous materials, and so on. In
recent years, a good amount of work has been reported in the area of steady [1-5] and
unsteady [6-12] combined conduction-radiation problems in an absorbing, emitting, and
scattering medium. Although most of the steady state studies were associated with
temperature boundary conditions, some of the papers discussed the problems with flux
boundary conditions. Tan and Lallemand [13] examined the transient temperature distribution
in a glass plate subjected to various boundary conditions. The transient heating of an
absorbing, emitting, and scattering material with different black plate boundary conditions
was studied by Yuen and Khatami [14]. They used a semi-explicit finite-difference scheme to
generate numerical solutions. For solving the transient energy equation, Tsai and Lin [15]
used the Crank-Nicholson scheme and the radiative part was solved by the nodal
3
approximation technique. Tong et al. [16] analyzed transient radiation heat transfer through
semi-isotropic scattering planar porous materials. They solved the radiative part using a two-
flux model.
The transient cooling problem with isotropic scattering in a cylindrical geometry was
analyzed by Baek et al. [17]. For solving the radiative part, they used the discrete ordinate
method, and a finite difference scheme was used to solve the transient energy equation. Tsai
and Nixon [18] considered a multilayered geometry without the effect of isotropic scattering.
Hahn et al. [19] examined the transient heat transfer using a multiflux model to solve the
radiative part. They analyzed the transient state of heat transfer in a layer of ceramic powder
during laser flash measurements of thermal diffusivity. Here also the effect of scattering was
not included. The problem with the melting of a semitransparent slab by an external radiating
source was investigated by Diaz and Viskanta [20]. A similar problem was also studied by
Seki and Nishimura et al. [21] Matthews et al. [22] discussed the steady and unsteady
combined conduction-radiation heat transfer heated by a highly concentrated solar radiation.
A two-flux model was used to solve the radiative part. Transient cooling of a semitransparent
material was examined by Siegel [10]. A finite difference procedure was adopted to obtain a
highly accurate temperature distribution across the layer. Yao and Chung [11] solved a
similar type of problem using an implicit finite volume scheme. The integral equation for
radiative heat flux was solved by a singularity subtraction technique and Gaussian
quadrature. Recently, studies related to this subject have been reviewed in detail by Siegel.
Although in the past decades a variety of computational schemes have been developed
to obtain an approximate solution to RTE, each scheme has demerits as well as merits in its
application. Techniques formerly used to solve RTE include the Eddington and Schuster-
Schwarzchild [23]. Even if the Monte Carlo [24] and zone methods [25] are called exact
numerical solutions, the former consumes an excessively large computational time and the
latter is not easily applicable to analyzing a radiatively scattering medium. Furthermore, most
of the methods previously adopted to solve RTE are incompatible with the finite difference
algorithm used in solving the continuity, momentum and energy equations which are
involved in various thermal and fluid mechanical problems
Numerical computations of radiation based on conventional methods such as the
zonal method, the Monte Carlo method (MCM), etc., prove to be laborious and very time
consuming. For this reason, numerous investigations [2–4] are currently being carried out
worldwide to assess computationally efficient methods.
4
Numerical simulations of the high-temperature appliances mentioned above require
multi-dimensional analysis. Treatment of radiative transport in the multidimensional
geometry is difficult mainly because of the three extra independent variables namely the
polar angle, the azimuthal angle and the wavelength. Since there is no way out to get rid of
the physical dimensions of the geometry and the wavelength of radiation, all numerical
models, except the zonal method and the MCM, deal with different types of discretization
schemes to make radiation less and less dependent on angular dimensions. The various
methods differ primarily in the angular discretization schemes and the use of either the
differential form or the integral form of the radiative transfer equation. The finite element
method for the calculation of a coupled conductive and radiative heat transfer problem in
two-dimensional rectangular enclosures [26]. However it was found to be not only time
consuming, but also difficult to apply to a scattering medium.
The main objective behind the development of any method for the solution of
radiative transport problems, apart from its versatility for various geometries, complex
medium conditions, etc., is that the method should be computationally efficient.
The finite volume method (FVM) for radiation is a robust method. It has many
advantages over other numerical methods such as the PN approximation, discrete transfer
method and the discrete ordinate method.
The present study examines the FVM for a coupled conductive and radiative heat
transfer in a two-dimensional square enclosure. In this work the conductive term is
discretized using the central differencing scheme and FVM approximation is adopted to
model the term of divergence of radiative heat flux in the energy equation. The solutions will
be presented using isothermal contours, radiative and total heat fluxes and transient behavior
of temperature for easy understanding of the phenomena involved.
CHAPTER 2
DISCRETIZATION METHODS BY USING FVM
6
2. DISCRETIZATION METHODS BY USING FVM:
2.1 THE NATURE OF NUMERICAL METHODS:
A numerical solution of a differential consists of a set of numbers from which the distribution
of the dependent variable φ can be constructed.
Let us suppose that we decide to represent the variation of φ by a polynomial in x,
φ = mm xaxaxaa ++++ ..........2
210 (2.1)
and employ a numerical method to find the finite number of coefficients .,.....,,, 210 maaaa
This will enable us to evaluate φ at any location x by substituting the value of x and the
values of the a’s into equation (2.1). This procedure is, however, somewhat inconvenient if
our ultimate interest is to obtain the value of φ at various locations.
Therefore, we should think for a numerical method, which treats as its basic
unknowns the values of the dependent variable at a finite number of locations (called the grid
points) in the calculation domain.
2.1.1 The Discretization Concept:
Now we have replaced the continuous information contained in the exact solution of the
differential equation with discrete values by focusing our attention on the values at the grid
points. We have thus discretized the distribution ofφ, and it is appropriate to refer to this class
of numerical methods as discretization methods.
The algebraic equations involving the unknown values of φ at chosen grid points,
which we shall now name the discretization equations, are derived from the differential
equation governingφ. In this derivation, we must employ some assumption about how φ
varies between the grid points. Although this “profile” of φ could be chosen such that a single
algebraic expression suffices for the whole calculation domain, it is often more practical to
use piecewise profile such that a given segment describes the variation of φ over only a small
region in terms of φ values at the grid points within and around that region. Thus, it is
common to subdivide the calculation domain into a number of sub domains or elements such
that a separate profile assumption can be associated with each sub domain.
7
In the discretization concept the continuum calculation domain has been discretized. It
is this systematic discretization of space and of the dependent variables that makes it possible
to replace the governing differential equation with simple algebraic equations, which can be
solved with relative ease.
2.1.2 The Structure of the Discretization Equation:
A discretization equation is an algebraic relation connecting the values of φ for a group of
grid points. Such an equation is derived from the differential equation governing φ and thus
expresses the same physical information as the differential equation. That only a few grid
points participate in a given discretization equation is a consequence of the piecewise nature
of the profiles chosen. The value of φ at a grid point thereby influences the distribution of φ
only in its immediate neighborhood. As the number of grid points becomes very large, the
solution of the discretization equations is expected to approach the exact solution of the
corresponding differential equation. This follows from the consideration that, as the grid
points get closer together, the change in φ between neighboring grid points becomes small,
and then the actual details of the profile assumption become unimportant.
2.2 METHODS OF DERIVING THE DISCRETIZATION EQUATIONS:
For a given differential equation, the required discretization equations can be derived in many
ways and Taylor-Series formulation is one of them.
Taylor-Series Formulation:
The usual procedure for deriving finite-difference equation consists of approximating the
derivatives in the differential equation via a truncated Taylor series. Let us consider the grid
points in fig (2.1). For grid point 2, located midway between grid points 1 and 3 such that
Δx = x2 - x1 = x3-x2, the Taylor-series expansion around 2 gives
( ) .....................21
22
22
221 −⎟⎟
⎠
⎞⎜⎜⎝
⎛Δ+⎟⎟
⎠
⎞⎜⎜⎝
⎛Δ−=
dxdx
dxdx φφφφ (2.2)
8
( ) ........21
22
22
223 +⎟⎟
⎠
⎞⎜⎜⎝
⎛Δ+⎟
⎠⎞
⎜⎝⎛Δ+=
dxdx
dxdx φφφφ (2.3)
Truncating the series just after the third term, and adding and subtracting the two equations,
we obtain
xdxd
Δ−
=⎟⎠⎞
⎜⎝⎛
213
2
φφφ (2.4)
( )2321
22
2 2xdx
dΔ
+−=⎟⎟
⎠
⎞⎜⎜⎝
⎛ φφφφ (2.5)
The substitution of such expression into the differential equation leads to the finite-difference
equation.
The method includes the assumption that the variation of φ is somewhat like a
polynomial in x, so that the higher derivatives are unimportant. This assumption, however,
leads to an undesirable formulation when, for example, exponential variations are
encountered.
1 2 3
∆x ∆x
x
Figure 2.1 Three successive grid points used for the Taylor-series expansion.
2.3 CONTROL-VOLUME FORMULATION
The Finite Volume Method (FVM) is one of the most versatile discretization techniques used
in CFD. Based on the control volume formulation of analytical fluid dynamics, the first step
in the FVM is to divide the domain into a number of control volumes (aka cells, elements)
where the variable of interest is located at the centroid of the control volume. The next step is
to integrate the differential form of the governing equations (very similar to the control
volume approach) over each control volume. Interpolation profiles are then assumed in order
to describe the variation of the concerned variable between cell centroids. The resulting
equation is called the discretized or discretization equation. In this manner, the discretization
equation expresses the conservation principle for the variable inside the control volume.
9
The most attractive features of the control-volume formulation is that the resulting
solution would imply that the integral conservation of quantities such as mass, momentum,
and energy is exactly satisfied over any group of control volumes and over the whole
calculation domain. Even a coarse grid solution exhibits exact integral balances.
Advantages: (1) Basic FV control volume balance does not limit cell shape.
(2)Mass, Momentum, Energy conserved even on coarse grids.
(3)Efficient, iterative solvers well developed.
(4) FVM enjoys an advantage in memory use and speed for very large problems, higher
speed flows, turbulent flows, and source term dominated flows (like combustion).
2.3.1 Finite Volume: Basic Methodology
(1)Divide the domain into control volumes.
(2)Integrate the differential equation over the control volume and apply the divergence
theorem.
(3)To evaluate derivative terms, values at the control volume faces are needed: have to make
an assumption about how the value varies.
(4)Result is a set of linear algebraic equations: one for each control volume.
(5)Solve iteratively or simultaneously.
2.3.2 Cells and Nodes
(1)Using finite volume method, the solution domain is subdivided into a finite number of
small control volumes (cells) by a grid.
(2)The grid defines the boundaries of the control volumes while the computational node lies
at the center of the control volume.
(3)The advantage of FVM is that the integral conservation is satisfied exactly over the control
volume.
Figure 2
Typical
(1)The
control
(2)The
by inter
Figure 2
2.2 Cells an
l Control vo
net flux th
volume fac
value of th
rpolation.
2.3 Typical
nd Nodes of
olume
hrough the c
ces (six in 3
he integrand
Control Vo
f Control Vo
control volu
D). The con
d is not avai
olume
10
olume
ume bound
ntrol volum
ilable at the
ary is the s
mes do not ov
e control vo
sum of inte
verlap.
olume faces
egrals over
s and is det
the four
termined
11
2.4 FINITE VOLUME METHOD FOR UNSTEADY FLOWS:
2.4.1 ONE-DIMENSIONAL UNSTEADY HEAT CONDUCTION:
Unsteady one-dimensional heat conduction is governed by the equation
SxTK
xtTc +
∂∂
∂∂
=∂∂ )(ρ
(2.6)
δxwP δxPe
δxwe
P W E w
e
Figure 2.4 One dimensional control volume
Consider the one-dimensional control volume in Figure-2.4 Integration of equation over the
control volume and over a time interval from t to t+Δt gives
dtsdVdVdtxTK
xdVdt
tTc
tt
t CV
tt
t CV
tt
t CV∫ ∫∫ ∫∫ ∫Δ+Δ+Δ+
+∂∂
∂∂
=∂∂ )(ρ
(2.7)
This may be written as
VdtSdtxTKA
xTKAdVdt
tTc
tt
t
tt
t we
e
w
tt
t
Δ+⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
∂∂
−⎟⎠⎞
⎜⎝⎛
∂∂
=⎥⎦
⎤⎢⎣
⎡
∂∂
∫∫∫ ∫Δ+Δ+Δ+
ρ (2.8)
In equation A is the face area of the control volume,ΔV is its volume, which is equal to AΔx
where Δx is the width of the control volume and is the average source strength.If the
temperature at a node is assumed to prevail over the whole control volume,the left hand side
can be written as
( ) VTTcdVdttTc o
PPCV
tt
t
Δ−=⎥⎦
⎤⎢⎣
⎡
∂∂
∫ ∫Δ+
ρρ (2.9)
In equation (2.9) superscript ‘o’ refers to temperature at time t; temperatures at time level
t+Δt are not superscripted. The same result as (2.9) would be obtained by substituting
( )t
TT oPP
Δ− for so this term has been discretised using a first (backward) differencing
12
scheme. If we apply central differencing schemes to the diffusion terms on the right hand side
equation (2.8) may be written as
( ) VdtSdtx
TTAKx
TTAKVTTctt
t
tt
t WP
WPw
PE
PEe
oPP Δ+⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −−⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=Δ− ∫∫
Δ+Δ+
δδρ (2.10)
To evaluate the right hand side of this equation we need to make an assumption about the
variation of PT , ET and WT with time. We could use temperatures at time t or at time t+Δt to
calculate the time integral or, alternatively, a combination of temperatures at time t+Δt. We
may generalize the approach by means of a weighing parameter θ between 0 and 1 and write
the integral TI of temperature PT with respect to time as
( )[ ] tTTdtTI oPP
tt
tPT Δ−+== ∫
Δ+
θθ 1
(2.11)
Hence
θ 0 1/2 1
TI tTP Δ0 ( ) tTT oPP Δ+2
1 tTPΔ
If θ=0 the temperature at (old) time level t is used; if θ=1 the temperature at new time level
t+Δt is used; and finally if θ=1/2, the temperature at t and t+Δt are equally weighed.
Using formula (2.11) for TW and TE in equation (2.10), and dividing by AΔt throughout, we
have
( ) ( )⎥⎦
⎤⎢⎣
⎡ −−
−=Δ⎟
⎟⎠
⎞⎜⎜⎝
⎛
Δ−
WP
WPw
PE
PEeo
PP
xTTK
xTTK
xtTT
cδδ
θρ
( ) ( ) ( )
xSx
TTKx
TTK
WP
oW
oPw
PE
oP
oEe Δ+
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−
−−+
δδθ1
(2.12)
13
This may be re-arranged to give
( )[ ] ( )[ ]+−++−+=⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛++
ΔΔ o
WWWP
woEE
PE
eP
WP
w
PE
e TTxK
TTxK
TxK
xK
txc θθ
δθθ
δδδθρ 11
( ) ( ) xSTxK
xK
txc o
PWP
w
PE
e Δ+⎥⎦
⎤⎢⎣
⎡−−−−
ΔΔ
δθ
δθρ 11 (2.13)
Now we identify the coefficients of TW and TE as aW and aE and write equation (2.13) in the
familiar standard form:
( )[ ] ( )[ ] ( ) ( )[ ] bTaaaTTaTTaTa oPEW
oP
oEEE
oWWWPP +−−−−+−++−+= θθθθθθ 1111 (2.14)
Where ( ) oPEWP aaaa ++= θ
And txca o
P ΔΔ
= ρ
With
aW aE b
WP
w
xKδ
PE
e
xKδ
∆
The exact form of the final discretised equation depends on the value of θ. When θ is zero,
we only use temperatures oE
oW
oP andTTT , at the old time level t on the right hand side of
equation (2.14) to evaluate TP at the new time; the resulting scheme is called explicit. When
0<θ<1 temperatures at the new time level are used on the both sides of the equation; the
resulting schemes are called implicit. The extreme case of θ=1 is termed fully implicit and
the case corresponding to θ=1/2 is called the Crank-Nicolson scheme.
2.4.2 EXPLICIT SCHEME:
In the explicit scheme the source term is linearised as b=Su +SpTpo. Now the substitution of
θ=0 into (2.14) gives the explicit discretisation of the unsteady conductive heat transfer
equation:
14
[ ] uo
PpEWo
Po
EEo
WWPP STSaaaTaTaTa +−+−++= )( (2.15)
PE
eE
WP
wW
oP
oPP
xK
a
xK
a
txca
aa
δ
δ
ρ
=
=
ΔΔ
=
=
The right side of equation (2.15) only contains values at the old time step so the left hand side
can be calculated by forward marching in time. The scheme is based on backward
differencing and its Taylor series truncation error accuracy is first-order with respect to time.
All the coefficients need to be positive in the discretised equation. The coefficients of oPT
may be viewed as the neighbor coefficient connecting the values at the old time level to those
at the new time level. For this coefficient to be positive we must have EWo
P aaa −− >0. For
constant K and uniform grid spacing, xxx WPPE Δ== δδ , this may be written as
xK
txc
ΔΔΔ 2
fρ or we can write
( )Kxct
2
2ΔΔ ρp
This inequality sets a stringent maximum limit to the time step size and represents a serious
limitation for the explicit scheme. It becomes very expensive to improve spatial accuracy
because the maximum possible time step needs to be reduced as the square of Δx.
Consequently, this method is not recommended for general transient problems.
2.4.3 THE FULLY IMPLICIT SCHEME:
When the value of θ is set equal to 1 we obtain the fully implicit scheme. The discretised
equation is
uo
Po
PEEWWPP STaTaTaTa +++= (2.16)
15
Where PEWo
PP Saaaa −++=
and txca o
P ΔΔ
= ρ
with
PE
eE
WP
wW
xK
a
xK
a
δ
δ
=
=
Both sides of the equation contain temperatures at the new time step, and a system of
algebraic equation must be solved at each time step. The time marching procedure starts with
a given initial field of temperatures To. The system of equations (2.16) is solved after
selecting time step Δt. Next the solution T is assigned to To and the procedure is repeated to
progress the solution by a further time step.
It can be seen that all the coefficients are positive, which makes the implicit scheme
unconditionally stable for any size of time step. The implicit method is recommended for
general purpose transient calculation because of its robustness and unconditional stability.
2.4.4 CRANK-NICOLSON SCHEME:
The Crank-Nicolson scheme method results from setting θ=1/2 in equation (2.14) . Now the
discretised unsteady heat conduction equation is
bTaaa
TTaTTaTa o
PWEo
p
oWW
W
oEE
EPP +⎥⎦⎤
⎢⎣⎡ −−+
⎥⎥⎦
⎤
⎢⎢⎣
⎡ ++⎥
⎦
⎤⎢⎣
⎡ +=
2222 (2.17)
Where ( ) Po
PEWP Saaaa21
21
−++=
And txca o
P ΔΔ
= ρ
aW aE b
WP
w
xKδ
PE
e
xKδ
oPPu TSS
21
+
16
Since more than one unknown value of T at the new time level is present in equation (2.17)
the method is implicit and simultaneous equations for all node points need to be solved at
each time step. Although schemes with 1/2 θ 1, including the Crank-Nicolson scheme, are
unconditionally stable for all values of the time step. It is more important to ensure that all the
coefficients are positive for physically realistic and bounded results. This is the case if the
coefficients of satisfies the following condition:
⎥⎦⎤
⎢⎣⎡ +
2WEo
Paa
a f
Which leads to
Kxct
2ΔΔ ρp
This time step limitation is only slightly less restrictive than associated with the explicit
method. The Crank-Nicolson method is based on central differencing and hence it is the
second-order accurate in time. With sufficiently small time steps it is possible to achieve
considerably greater accuracy than with the explicit method. The overall accuracy of a
computation depends upon on the spatial differencing practice, so the Crank-Nicolson
scheme is normally used in conjunction with spatial central differencing.
2.5 TWO-DIMENSIONAL UNSTEADY HEAT CONDUCTION:
A portion of a two-dimensional grid is shown in fig-2.5.For the grid point P, points E and W
are its x-direction neighbors, while N and S (denoting north and south) are the y-direction
neighbors. The control volume around P is shown by dashed lines. Its thickness in z direction
is assumed to be unity. The actual location of the control volume faces in relation to the grid
points is exactly midway between the neighboring grid points.
SyTK
yxTK
xtTc +
∂∂
∂∂
+∂∂
∂∂
=∂∂ )()(ρ
(2.18)
Figure 2
We can
assume
through
equatio
tt
t CV∫ ∫Δ+
ρ
This ma
e
w
tt
t∫ ∫
Δ+
⎢⎣
⎡ρ
In equa
where Δ
AΔy, w
2.5 Two Di
n calculate
that qe, thu
h the other f
n can be tur
dVdttTc =∂∂ρ
ay be writte
dVdttTc ⎥
⎦
⎤
∂∂ρ
ation A is th
Δx is the w
where Δy is
mensional C
the heat fl
us obtained
faces can b
rned into th
Kx
tt
t CV∫ ∫Δ+
∂∂
= (
en as
KAVtt
t∫Δ+
⎢⎣
⎡⎜⎝⎛=
he face area
idth of the
s the width
Control Vol
ux qe at th
d, prevails o
be obtained
e discretiza
dVdtxTK +∂∂ )
KxTA
e⎜⎝⎛−⎟
⎠⎞
∂∂
a of the cont
control vol
of the con
17
lume
he control-v
over the en
in a simila
ation equatio
x
tt
t CV∫ ∫Δ+
⎜⎜⎝
⎛∂∂
+
dtxTKA
w⎥⎦
⎤⎟⎠⎞
∂∂
trol volume
ume in x di
ntrol volume
volume face
ntire face of
ar fashion. I
on
yTK
t
t∫Δ+
+⎟⎟⎠
⎞∂∂
KAtt
t∫Δ+
⎢⎢⎣
⎡⎜⎜⎝
⎛+
VdS
tt
t
Δ+ ∫Δ+
e,ΔV is its v
irection and
e in y direc
e between P
f area Δy x
In this mann
dtsdVt
CV∫ ∫Δ
KAyT
n⎜⎜⎝
⎛−⎟⎟
⎠
⎞∂∂
dt
volume, whi
d in y direc
ction. is
P and E. W
x 1.Heat flo
ner, the dif
dtyTKA
s ⎥⎥⎦
⎤⎟⎟⎠
⎞∂∂
ich is equal
tion ΔV is
the average
We shall
ow rates
fferential
(2.19)
(2.20)
l to AΔx
equal to
e source
18
strength. If the temperature at a node is assumed to prevail over the whole control volume,
the left hand side can be written as
( ) VTTcdVdttTc o
PPCV
tt
t
Δ−=⎥⎦
⎤⎢⎣
⎡
∂∂
∫ ∫Δ+
ρρ
(2.21)
In equation (2.21) superscript ‘o’ refers to temperature at time t; temperatures at time level
t+Δt are not superscripted. The same result as (2.21) would be obtained by substituting
( )t
TT oPP
Δ− for so this term has been discretised using a first (backward) differencing
scheme. If we apply central differencing schemes to the diffusion terms on the right hand side
equation (2.20) may be written as
( ) dtx
TTAKx
TTAKVTTctt
t WP
WPww
PE
PEee
oPP ∫
Δ+
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −−⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=Δ−
δδρ
+ VdtSdty
TTAK
yTT
AKtt
t
tt
t SP
SPss
PN
PNnn Δ+⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −−⎟⎟
⎠
⎞⎜⎜⎝
⎛ −∫∫Δ+Δ+
δδ (2.22)
To evaluate the right hand side of this equation we need to make an assumption about the
variation of PT , ET , WT ,TN, and TS with time. We could use temperatures at time t or at time
t+Δt to calculate the time integral or, alternatively, a combination of temperatures at time
t+Δt. We may generalize the approach by means of a weighing parameter θ between 0 and 1
and write the integral TI of temperature PT with respect to time as
( )[ ] tTTdtTI oPP
tt
tPT Δ−+== ∫
Δ+
θθ 1
(2.23)
Hence
θ 0 1/2 1
TI tTP Δ0 ( ) tTT oPP Δ+2
1 tTPΔ
If θ=0 the temperature at (old) time level t is used; if θ=1 the temperature at new time level
t+Δt is used; and finally if θ=1/2, the temperature at t and t+Δt are equally weighed.
19
Using formula for TW and TE and TS and TN in equation, and dividing by Δt throughout, we
have
( ) ( )⎥⎦
⎤⎢⎣
⎡ −−
−=ΔΔ⎟
⎟⎠
⎞⎜⎜⎝
⎛
Δ−
WP
WPww
PE
PEeeo
PP
xTTAK
xTTAKyx
tTTc
δδθρ
( ) ( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−
−−+
WP
oW
oPww
PE
oP
oEee
xTTAK
xTTAK
δδθ1
+ ( ) ( )⎥⎦
⎤⎢⎣
⎡ −−
−
SP
SPss
PN
PNnn
yTTAK
yTTAK
δδθ
( ) ( )
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−
−−+
SP
oS
oPss
PN
oP
oNnn
yTTAK
yTTAK
δδθ )1( + yxS ΔΔ (2.24)
This may be rearranged to give
( )[ ] ( )( )oWW
WP
wwoEE
PE
eeP
SP
ss
PN
nn
WP
ww
PE
ee TTx
AKTT
xAK
Ty
AKy
AKx
AKx
AKtyxc θθ
δθθ
δδδδδθρ −++−+=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛++++
ΔΔΔ 11
+ ( )[ ] ( )[ ]oSS
SP
ssoNN
PN
nn TTy
AKTT
yAK
θθδ
θθδ
−++−+ 11
+ ( ) ( ) ( ) ( ) yxSTy
AKy
AKx
AKx
AKtyxc o
PSP
ss
PN
nn
WP
ww
PE
ee ΔΔ+⎥⎦
⎤⎢⎣
⎡−−−−−−−−
ΔΔΔ
δθ
δθ
δθ
δθρ 1111
(2.25)
( )[ ] ( )[ ] ( )[ ] ( )[ ]oNNN
oSSS
oEEE
oWWWPP TTaTTaTTaTTaTa θθθθθθθθ −++−++−++−+= 1111
+ ( ) ( ) ( ) ( )[ ] yxSTaaaaa oPNSEW
oP ΔΔ+−−−−−−−− θθθθ 1111 (2.26)
)( NSEWo
PP aaaaaa ++++= θ
For implicit scheme the value of θ is 1.The equation can be written as
aPTP =aETE + aWTW + aNTN + aSTS + b (2.27)
tyxca o
P ΔΔΔ
= ρ
20
oP
oPC TayxSb +ΔΔ=
yxSaaaaaa Po
PSNWEP ΔΔ−++++=
The product Δx Δy is the control volume.
1 D 2 D
ΔV Δx ΔxΔy
Aw=Ae 1 Δy
An=As --- Δx
aW aE aS aN
2D
WP
ww
xAK
δ
PE
ee
xAK
δ
SP
ss
yAK
δ
PN
nn
yAK
δ
2.6 Solutions of Algebraic Equations:
Direct Methods (i.e., those requiring no iteration) for solving the algebraic equations arising
in two or three-dimensional problems are much more complicated and require rather large
amounts of computer storage and time. For a linear problem, which requires the solution of
the algebraic equations only once, a direct method may be acceptable; but in nonlinear
problems, since the equations have to be solved repeatedly with updated coefficients, the use
of a direct method usually not economical. We shall, therefore exclude direct methods from
further consideration.
The alternative method then is iterative methods for the solution of algebraic
equations. These start from a guessed field of T (the dependent variable) and use the algebraic
equations in some manner to obtain an improved field. Successive repetitions of the
algorithm finally lead to a solution that is sufficiently close to the correct solution of the
algebraic equations. Iterative methods usually require very small additional storage in the
21
computer, because only non-zero coefficients of the equations are stored in core memory. So
they are especially useful for handling nonlinearities.
There are many iterative methods for solving algebraic equations. Some of the
examples are the Jacobi and Gauss-Seidel point-by-point iteration methods.
Jacobi and Gauss-Seidel iterative methods are easy to implement in simple computer
programs, but they can be slow to converge when the system of equations is large. Hence
they are not considered suitable for rapidly solving tri-diagonal systems that is now called
Thomas algorithm or the tri-diagonal matrix algorithm (TDMA). The TDMA is actually a
direct method for one-dimensional situations, but it can be applied iteratively, in a line-by-
line fashion, to solve multi-dimensional problems. It is computationally inexpensive and has
the advantage that it requires a minimum amount of storage.
2.6.1 The Tri-diagonal Matrix Algorithm:
TDMA refers to the fact that when the matrix of the coefficients of these equations is
written, all the nonzero coefficients align themselves along three diagonals of the matrix.
Suppose the grid points were numbered 1, 2, 3, ……, N , with points 1 and N denoting
the boundary points. The discretization equations can be written as
iiiiiii dTcTbTa ++= −+ 11 (2.28)
for i=1, 2, 3, ……..,N. Thus the temperature Ti is related to the neighboring temperatures Ti+1
and Ti-1. To account for the special form of the boundary-point equations, let us set
c1=0 and bN =0,
so that the temperatures T0 and TN+1 will not have any meaningful role to play. (When the
boundary temperatures are given, these boundary-point equations take a rather trivial form.
For example, if T1 is given, we have a1=1, b1=0, c1=0, and d1=the given value of T1. )
These conditions imply that T1 is known in terms of T2. The equation for i=2 is a
relation between T1, T2, and T3. But, since T1 can be expressed in terms of T2, this relation
reduces to a relation between T2 andT3. In other words, T2 can be expressed in terms of T3.
This process of substitution can be continued until TN is formally expressed in terms of TN+1.
But, because TN+1 have no meaningful existence, we actually obtain the numerical value of
TN at this stage. This enables us to begin the “back-substitution” process in which TN-1 is
22
obtained from TN, TN-2 from TN-1,……,T2 from T3, and T1 from T2. This is the essence of
TDMA. Suppose in the forward- substitution process, we seek a relation
Ti=PiTi+1+Qi (2.29)
Similarly we can also write
Ti-1= Pi-1Ti +Qi-1 (2.30)
Substitution of equation (2.30) into equation (2.28) leads to
aiTi=biTi+1+ci(Pi-1Ti+Qi-1) + di (2.31)
The coefficients Pi and Qi are
1
1
1
,
−
−
−
−+
=
−=
iii
iiii
iii
ii
PcaQcd
Q
Pcab
P
These are the recurrence relations, since they give Pi and Qi in terms of Pi-1 and Qi-1. To start
the Recurrence process, we note that equation (2.28) for i=1 is almost of the form (2.30).
Thus the values of P1 Q1 are given by
1
11
1
11
ad
Q
ab
P
=
=
Discussion: (1) The discretization equations for the grid points along a chosen line are
considered. They contain the temperature at the grid points along the neighboring lines. If
these temperatures are substituted from their latest values, the equation for the grid points
along the chosen line would look like one-dimensional equations and could be solved by the
TDMA. This procedure is carried out for all the lines in the y-direction and may be followed
by a similar treatment for the x-direction.
(2) The convergence of the line-by-line method is faster, because the boundary-condition
information from the ends of the line is transmitted at once to the interior of the domain, no
matter how many grid points lie in one line.
2.7 Com
2.8 Flow
300oC
mputationa
Fi
w Chart F
al Domain:
igure 2.6 Co
For Solving
L
omputationa
Transient
III
600oC
300oC
INITIA
23
al Domain
Conductio
ALIZATION
L
on:
OF TEMPER
L 300oC
RATURE
CHAPTER 3
RADIATIVE HEAT TRANSFER
25
3. RADIATIVE HEAT TRANSFER: 3.1 Mathematical Description of Radiation:
We refer as radiative energy transfer to the variation of the energy of any system due to
absorption or emission of electromagnetic waves. From a physical point of view, such
electromagnetic waves can be understood as a group of mass less particles that propagate at
the speed of light ‘c’. Each of these particles carries an amount of energy, inversely
proportional to the wavelength of its associated wave. Such particles are photons. A single
photon represents a plane wave; therefore we are forced to assume that it propagates along a
straight path. Because of this, we can think of a system loosing energy by emitting a number
of photons, or gaining energy by absorbing them.
Radiation heat transfer presents a number of unique characteristics. First of all, the
amount of radiative heat transfer does not depend on linear differences of temperature, but on
the difference of the fourth power of the temperature. This fact implies that, at high
temperatures, radiative energy transfer should be taken into account, particularly if large
differences in temperature exist. Furthermore, it is the only one form of energy transfer in
vacuum. Therefore, in vacuum applications, radiation should be taken into account, even low
temperatures. And, of course, it should be taken into account for study of devices which use
solar energy.
On the other hand, radiation can be neglected if there are no significant temperature
differences in a given system, and also if highly reflective walls are present.
As the photons propagate along straight paths, we also need to know, at every spatial
location, the number of photons propagating in a given solid angle. Under the assumption of
straight propagation, we are neglecting the wave properties of radiation.
The photon density is defined as the number of photons, with wavelength between λ
and λ + dλ, crossing an area dA perpendicular to the photon direction s within a solid angle
dΩ, per unit time dt and per unit wavelength dλ. It is more useful to consider the energy
density I instead of the photon density. The energy density is simply the number of photons
multiplied by the energy of each photon. The energy density is usually named intensity
radiation field, and its physical meaning is therefore
=),ˆ,( λsrI s (3.1)
26
From this definition is clear that, for a fixed point and wavelength, the intensity radiation
field is a function defined over the unit sphere, and therefore is not a conventional scalar
field, such as the temperature. With the definition of the intensity radiation field on mind, the
total energy that crosses a surface perpendicular to a given unit vector n , per unit time and
area, is readily obtained as
),,().(),(0 4
λλπ
∧∞ ∧∧
∫ ∫ Ω= srIsnddnrq ) (3.2)
Where ⎟⎠⎞
⎜⎝⎛ ∧∧
sn. factor appears since I is defined as energy through the propagation direction∧
s .
3.2 Radiative Properties of Materials:
3.2.1 Black body radiation:
Once the mathematical tools for the description of radiation energy are settled, it
would be interesting to know how much energy emits a body under different physical
conditions. This one turns out to be a complicated problem, so some simplifying hypothesis is
required. The simplest possible case is that of the black body.
The black body plays the same role as the ideal gas, in the sense that its behavior is the
same no matter of what is made. It also serves as a basis for more complicated bodies. By
definition, any body which absorbs the totality of photons is called a black body. For such a
body in thermal equilibrium with its surroundings, it has to emit as much radiation energy as
it absorbs, otherwise it will heat up or cool down. This fact is called the Kirchhoff law. The
energy emitted by a black body depends only on its temperature, and the distribution over all
wavelengths is
,1)exp(
12),( 5
2
−=
Thc
hcTIb
λσλ
λ
(3.3)
Where h is the Planck’s constant, σ is the Boltzmann constant, and c is the speed of light. The
photon emission by a black body is isotropic, that is, it does not depend on any particular
direction s . Therefore, by using equation (3.2), carrying out the angular integration over an
hemisphere, we find that the energy emitted by a black body per unit area, time and
wavelength is simply ),(),( λπλ TITE bb = .
27
An important feature of a black body is the emissive power per unit area, which is readily
found by integrating equation (3.3) over all wavelengths. The result is the well known Stefan-
Boltzmann law, which relates the emissive power to the temperature of the black body:
πσ 4
)( TTI Bb =
(3.4)
3.2.2 Surface properties:
Each surface will radiate a certain amount of energy due to its temperature. Such emitted
energy, which will depend on the properties of the surface, its temperature, the radiation
wavelength, the direction of emission leads to the definition of the emissivity of a surface. As
stated before, we can use the black body as a reference to define the surface properties of real
bodies. Therefore, if the surface is at temperature T, the emissivity is defined as
),()ˆ,,(
λλ
εTI
sTI
b
emitted=
(3.5)
As the black body is defined as the perfect radiation absorber, by the Kirchhoff law, there is
no body capable of emitting more radiation energy than a black body for any given
temperature. Therefore, the emissivity ε will be between zero and one.
On the other hand, when an electromagnetic wave strikes a surface, the interaction
between this incident wave and the elements of the surface result on a fraction of the energy
of the wave being reflected, while the remaining fraction will be absorbed (or transmitted)
within the material. These fractions may very well depend on the incident angle and the
wavelength of the original wave. Specifically, we can define
ginco
reflected
II
min
=ρ ; ginco
absorbed
II
min
=α ; ginco
dtransmitte
II
min
=τ
The coefficientsρ,α, and τ, are the reflectivity, absorptivity, and transmissivity of the surface
respectively. As the wave is either reflected, or absorbed, or transmitted, it is clear that we
must have ρ +α +τ = 1 if the energy is to be conserved.
An important feature of reflection should be pointed out: while the reflected wave has
the same angle (with respect to the normal of the surface) of the incident wave for perfectly
smooth surfaces, if the surface is not polished, the angle of the reflected wave could have any
value, due to multiple reflections. There is a fraction of the incoming intensity that is equally
distributed for all possible outgoing directions. Due to this, the reflectivity ρ is usually
28
divided in a diffuse componentρd, and a specular component ρs. Therefore, a fraction ρd of
the incoming energy will be reflected equally over all the angles, and a fractionρs will be
reflected in an angle equal to that of the incident wave. We have of course ρ = ρd +ρs.
Moreover, a surface that absorbs a fraction α of the incident energy will emit the
same amount of energy if it is at thermal equilibrium (the Kirchhoff law again). Therefore,
we could assume that )ˆ,,()ˆ,,( sTsT λελα = .
3.3 FORMULATION OF RADIATIVE TRANSFER EQUATION BY
USING FV METHOD:
We are assuming that photons propagate along straight lines, hence the most natural way to
examine the effect exerted by the medium on the number of photons (or equivalently on the
intensity radiation field) is to analyze the intensity radiation field precisely along a straight
line. In the most general case, the intensity along the direction defined by the unit vector s
will depend both on the distance s in this direction, and on time t. Therefore we could write
the variation of intensity with respect to s and t as
( ) ( ) dttIds
sItsIdttdssIdI
∂∂
+∂∂
=−++= ,,
(3.6)
Recalling that photons travel at the speed of light c, we have ds = cdt, and the resulting
variation of intensity per unit length along the direction s is
tI
csI
dsdI
∂∂
+∂∂
=1
(3.7)
The speed of light is very high, resulting on the fact that, for practical purposes, we can think
that the intensity radiation field instantly reacts to any changes of the physical conditions that
determine it. Therefore, the partial derivative of I with respect to time t in equation, will be
ignored hereinafter.
As an electromagnetic wave propagates inside a medium, it loses energy as the charged
particles within the medium accelerate in response to the wave. These particles, in turn,
release part of its energy in form of electromagnetic waves. In the photon framework, we
think of the same process as photons being absorbed and emitted by the medium. However, if
the final state of the interacting particle is the same as the initial state, we understand that the
corresponding photon is simply redirected (and therefore has the same energy). We say that
29
such a photon is scattered. It turns out that scattered photons complicate the formulation of
the radiative transfer equation (RTE), by turning a simple linear differential equation to an
integro-differential one, combining differentiation with respect one set of variables (spatial
location) and integration over another set of variables (solid angle).
The RTE accounts for the variation of the intensity radiation field, readily related to the
number of photons, on a direction given by the unit vector s . Such variation can be attributed
to different phenomena, and is usually divided in three additive terms. It is written as
( ) ∫ Ω′′++−=π
φπ
σκβ4
')ˆ,ˆ()ˆ,(4
)()ˆ,()()ˆ,()ˆ,( dsssrIrsrIrsrIrds
srdIb (3.8)
The above equation is valid for a single wavelength. If the absorption and scattering coeffi-
cients κ andσs are zero, the RTE is simplified enormously. Under such conditions, we talk of
a transparent medium, and for very small domains, compared to 1/κ or 1/σs, this
approximation is reliable.
The first term, which is negative, accounts for the decrease of the number of photons on the
given direction, either because it is absorbed by the medium (with an absorption coefficientκ)
or because it is scattered onto another direction (with a scattering coefficient σs). The second
term has positive sign, therefore implying an increase of the number of photons. This term is
due to thermal emission of photons. It is zero only if the temperature is zero, or the
absorption coefficient is zero. Notice that the proportionality coefficient is the same
absorption coefficient appearing in the first term. This holds under the assumption of local
thermodynamic equilibrium.
The third term contributes only if the medium scatters radiation. We should take into account
that any photon, propagating along a direction given by s′ˆ , may be redirected to the analyzed
direction s . We define the function ( )ss ˆ,ˆ′φ to be 4π times the probability of such redirection
occurring. Then we integrate to consider all possible directions s′ˆ . The function ( )ss ˆ,ˆ′φ is
known as the phase function. It has to be normalized, in a way that
∫ =Ω ′′π
πφ4
4)ˆ,ˆ( dss
(3.9)
This equation merely states that the incoming photon should be scattered into some other
direction. Where the extinction coefficient and source function are
)()()( rrr σκβ += (3.10)
30
∫ Ω′Φ+=ππ
σκ4
)ˆ,ˆ()ˆ,(4
)()ˆ,()()ˆ,( dsssrIrsrIrsrS b
(3.11)
r is the position vector and s is the unit vector describing the radiation direction. Equation
(3.8) indicates the intensity depends on spatial position and angular direction. To discretize it
finite volume method is used. The control angle used here are the solid angle proposed and
used by Raithbay and coworkers[27-29]. Following the control volume spatial discretization
practice, the angular space is subdivided into Nθ X Nφ =M control angles in any desired
manner.
Figure 3.1Typical control angle
Figure 3.2 Intensity Il in direction in the center of the elemental sub-solid angle Ωl
lΔΩ
31
Integrating equation (3.8) over a typical two-dimensional Δv and ΔΩl gives
∫ ∫∫ ∫Ω ′Δ ΔΩ ′Δ Δ
Ω+−=Ω ′′
v
lll
v
dvdSIdvdds
Id )( β
(3.12)
where
).ˆ,( srII ′≡′
Applying the divergence theorem, Eq(3.12) becomes
∫ ∫ ∫∫ΔΩ ΔΩ ΔΔ
Ω+−=Ωl l v
llll
A
ll dvdSIdAdnsI )()ˆ.ˆ( β
(3.13)
The left side of the equation represents the inflow and outflow of radiant energy across the
four control volume faces. The right hand side denoted the attenuation and augmentation of
energy within a control volume. Following the practice of control volume approach, the
intensity is assumed constant within a control volume and a control angle. So Eq.(3.13) can
be simplified to
Ω ′ΔΔ+−=ΩΔ ∫∑ΔΩ=
vSIdnsAI llli
li
ii
l
l
)()ˆ.ˆ(4
1
β
(3.14)
Where
lllM
r
lsb IIS ′′
=
′ ΔΩΦ+=′ ∑14π
σκ
(3.15)
In Eq. (3.15), the radiation direction varies within a control angle, whereas the magnitude of
the intensity is assumed constant. If the radiation direction is fixed at a given direction within
a control angle and the magnitude of intensity is constant, the discretization equation for the
discrete ordinates method [30] is obtained.
Followi
modifie
With th
∑=
Ii
il
4
1
For the
simplifi
Figure 3
−leI(
To rela
needed.
intensit
ing the trea
ed source fu
his modifica
Δ ∫ΔΩ
sA li
l
.ˆ(
typical con
ied to
3.3 Control
Δ− cxl
w DAI )
ate the boun
. One avail
ies equal to
atment prese
unction can
mS
ation, Eq. (3
=Ωdn li )ˆ.
ntrol volum
angle orien
+lcx
ln II −(
ndary inten
lable schem
o the upstrea
lPI
ented by ch
be written f
lm =β
+= blm Iκ
.14) becom
+′− I lm( β
e and radiat
ntations
lcyy
ls DAI Δ)
nsities to th
me is the s
am nodal int
leP II ==
32
hai et al;[30]
for a discret
s Φ−π
σβ
4
∑≠=
M
ll
s
',1'4πσ
mes
ΔΔ′+ vS m )
tion directio
pl
m−= )([ β
he nodal int
step schem
tensities;
ls
ln III =,
] a modifie
te direction
lllΔΩ
′
≠
ΔΩΦl
lllI''
Ω ′Δ
on shown in
lmp
lp SI + )(
tensity, spa
e which se
lw
lS III =,
d extinction
l as
Ω l '
n Fig, Eq.(3
lp vΔΩΔ])
atial differe
ets the dow
lWI
n coefficien
3.16) can be
encing sche
wnstream b
nts and a
(3.16)
e further
(3.17)
mes are
oundary
33
From the above fig, the step scheme discretization equation can be written as
bIaIaIa l
SlS
lW
lW
lp
lp ++=
(3.18)
Where[ ]
lp
lm
lp
lm
lcx
lcy
lp
lcy
lS
lcx
lW
vSb
vyDxDa
xDa
yDa
ΔΩΔ=′
ΔΩΔ+Δ+Δ=
Δ=
Δ=
)(
)(β
∫ ∫ Φ=Ω ′Δ2
1
2
1
sinφ
φ
θ
θ
θθ dd
(3.19)
lcx
lcwx
lx
llcex DDdddnsD =−==Ω= ∫ ∫∫
2
1
2
1
sincossin)ˆ.ˆ(φ
φ
θ
θ
φθθφθ
(3.20)
lcy
lcsy
ly
llcny DDdddnsD =−==Ω= ∫ ∫∫
2
1
2
1
sinsinsin)ˆ.ˆ(φ
φ
θ
θ
φθθφθ
(3.21)
34
3.4 SOLUTION PROCEDURE:
Algorithm
are calculatedGuess unknown (i.e , )
Set the current intensities as guessed value
Calculate
baaa lp
lS
lW ,,,
lPI
If No
If Yes Record6' 10/ −≤− lp
lp
lp III
⎟⎟⎠
⎞⎜⎜⎝
⎛ ++= l
p
lS
lS
lW
lWl
p abIaIaI
lWIl
SIlnbI
lPI
Take Updated and lnbI l
PI
Figure 3.4 Flowchart for overall solution procedure
CHAPTER 4
ANALYSIS
36
4. ANALYSIS
4.1 ENERGY EQUATION
The transient state energy conservation equation consisting of both conduction and radiation in an infinitesimal control volume can be expressed as
RqTKtTc .2 ∇−∇=∂∂ρ (4.1)
It is assumed that the thermal conductivity K of the medium is independent of temperature T of the medium. The divergence of the radiative heat flux is given by
)4(. GIq bR −=∇ πκ (4.2)
Where κ is the absorption coefficient, Ib is the blackbody intensity and G is the incident radiation. Substituting Eq.(4.2) into Eq(4.1), the non dimensional equation can be expressed as
( )⎥⎦
⎤⎢⎣
⎡−
−−
∂∂
+∂∂
=∂∂
πθωθ
βθ
βξθ
4111 *
42
2
222
2
22G
NYHXL (4.3)
refTT
=θ , 34 refTKNσβ
= , πσ /4
*
refTGG =
Here in the above equation, Tref is the reference temperature, σ is the Stefan-Boltzmann constant, N is the conduction radiation parameter and β is the extinction coefficient of the medium.
4.2 Calculation of incident radiation and heat flux
For calculation of the divergence of radiative heat flux, incident radiation G distribution is required. It can be calculated as
∫ Ω=π4
)ˆ,()( dsrIrG rr
φθθ
π
φ
π
θddI∫ ∫
==
=2
00
sin (4.5)
37
The radiative heat flux in x, y and z directions can be calculated as
∫ Ω=π4
, )ˆ.ˆ)(ˆ,()( dessrIrq xxRrr
∫ ∫=
=π
φ
π
θ
φθθφθ2
0
sincossin ddI (4.6)
∫ Ω=π4
, )ˆ.ˆ)(ˆ,()( dessrIrq yyRrr
∫ ∫=
=π
φ
π
θ
φθθφθ2
0
sinsinsin ddI (4.7)
∫ Ω=π4
, )ˆ.ˆ)(ˆ,()( dessrIrq zzRrr
∫ ∫=
=π
φ
π
θ
φθθθ2
0
sincos ddI (4.8)
The total heat flux qT is the summation of both conductive and radiative heat fluxes. In non-dimensional form, total heat flux in any direction(e.g. x direction can be expressed as)
4,
4, 4
ref
xR
Xref
xT
T
qN
T
q
στθ
σ+
∂∂
−= (4.9)
Where τx (=βx) is the optical thickness of the medium in x-direction
CHAPTER 5
RESULTS AND DISCUSSION
39
In this section, the results obtained by solving the governing equation meant for coupled
conduction and radiation phenomena within an enclosure, have been delineated .In order to
sense the effect of radiative properties of the medium and surface emissivity, at first, pure
conduction phenomena has been discussed. The transient effect has also been discussed. In
the figure 5.1 the temperature distribution for steady state conduction for the computational
domain has been shown. The symmetry about X=0.5, is observed as expected. The heat flux
variations at bottom wall, top wall and side wall are shown in the fig. 5.2, fig.5.3, fig. 5.4, the
hot (bottom) wall compared to the portions nearing side walls. Where as the phenomenon is
reversed at the top cold wall. Heat flux at the bottom portion of the side walls is found to be
intensive in nature respectively. It is observed that less amount of heat gets transferred at the
central portion of the bottom wall.
Figure 5.1 Temperature isotherms at the steady state.
40
Figure 5.2 Variation of Heat flux at the bottom wall along X-axis
Figure 5.3 Variation of heat flux at the top cold wall with X axis
0
2
4
6
8
10
12
0 0.2 0.4 0.6 0.8 1
Dim
ension
less Heatflux
Dimensionless Coordinate X
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.2 0.4 0.6 0.8 1
Dim
ension
less Heat flu
x
Dimensionless Coordinate X
41
Figure 5.4 Variation of heat flux at the side cold wall with Y-axis
Before illustrating the effect of radiative properties of the participating medium on combined
conduction and radiation phenomena, the validation of the present code has been made
against the work Kim and Baek[17] (shown in the figure ).
Figures shows the temperature along the y-direction at the symmetry line X=0.5 for various
values of N. In addition ω=0 is assumed, which represents a non-scattering medium in which
the radiation is emitted and absorbed only. As N decreases, radiation plays a more significant
role than conduction. Therefore as N decreases, a steeper temperature gradient is formed at
both top and bottom walls and the medium temperature inside increases.
0
2
4
6
8
10
12
0 0.2 0.4 0.6 0.8 1
Dim
ension
less Heat flu
x
Dimensionless Coordinate Y
42
Figure 5.5 Dimensionless temperature profiles for N=1
Figure 5.6 Dimensionless temperature profiles for N=0.1
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Dim
ension
less Tem
perature θ
Dimensionless Coordinate Y
present
Kim and Baek
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Dim
ension
less Tem
perature θ
Dimensionless Coordinate Y
Present
Kim and Baek
43
Figure 5.7 Dimensionless temperature profiles for N=0.01
Figure 5.8 Dimensionless temperature profiles for N=0.001
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Dim
ension
less Tem
perature θ
Dimensionless Coordinate Y
Present
Kim and Baek
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Dim
ension
less Tem
perature θ
Dimensionless Coordinate Y
Present
Kim and Baek
44
5.3 Effect of Conduction- radiation parameter (N)
In general the fractional radiative heat flux is seen to decrease as N increases. In the hot wall
corner region, extremely large temperature gradient produces a great deal of conductive heat
flux as N increases. For N=0.001, QR/Q becomes nearly uniform along both end walls as
shown in the figure. In this study conduction radiation parameter N is varying, keeping εw =1,
and τl=1. Figure shows how fractional radiative heat flux changes with different values of
conduction radiation parameter in bottom hot wall and cold top wall.
Figure 5.9 Variation of fractional radiative heat flux for N=1;
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
Fraction
al ra
diative Heat flu
x
Dimensionalless Coordinte X
Bottom Hot Wall
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
Fraction
al ra
diative Heat flu
x
Dimensional Cooridnates X
Top Cold Wall
45
Figure 5.10 Variation of fractional radiative heat flux for N=0.1;
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
Fraction
al Rad
iative Heat Flux
Dimensionless Coordinate X
Bottom Hot Wall
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
Fraction
al Rad
iative Heat Fluxs
Dimensionless Coordinate X
Top Cold Wall"
46
Figure 5.11 Variation of fractional radiative heat flux for N=0.01;
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
Fraction
al Rad
iative Heat Fluxs
Dimensionless Coordinate X
Bottom Hot Wall
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
Fraction
al Rad
iative Hea
flux
Dimensionless Coordinate X
Top Cold Wall
47
Figure 5.12 Variation of fractional radiative heat flux for N=0.001;
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
Fraction
al Rad
iative Hea
flux
Dimensionless Coordinate X
Bottom Hot Wall
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
Fraction
al Rad
iative Hea
flux
Dimensionless Coordinate X
Top Cold Wall
48
5.4 Effect of wall emissivities at the hot bottom wall, keeping N=0.05, τl=1and ω=0.
As the wall emissivity increases, the intensity at the hot wall becomes strong and it further
increases the radiative heat flux and the temperature of the medium inside. The temperature
gradient in the vicinity of the hot wall is thus reduced by as much. The effect of emissivity is
found to be more pronounced for a thicker medium which has a higher optical thickness. The
total heat flux decreases with decrease in εw.
Figure 5.13 Variation of the total heat flux for εw=1 at the bottom wall, Y=0
Figure 5.14 Variation of the total heat flux for εw=0.8, at the bottom wall, Y=0.
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5
DIM
ONSIONlESS TOTA
L HEA
T FLUX
Dimensionless Coordinate X
N=0.05
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5
DIM
ONSIONlESS TOTA
L HEA
T FLUX
DIMENSIONLESS CORDINATES X
N=0.05
49
Figure 5.15 Variation of total heat flux for εw=0.5, at the bottom wall, Y=0.
Figure 5.16 Variation of total heat flux for εw=0.2, at the bottom wall, Y=0.
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5
DIM
ONSIONlESS TOTA
L HEA
T FLUX
DIMENSIONLESS CORDINATES X
N=0.05
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5
Dim
ension
al to
tal heat flux
Dimensionless Coordinate X
N=0.05
50
5.5 Temperature Isotherms for different scattering albedo and emissivity at steady
state conditions: The figures shown below are the temperature isotherms at steady state. In
all the cases the conduction-radiation parameter N=0.001 is taken, that makes the phenomena
as radiation dominant. Fig.5.17 shows the medium is non-scattering, therefore the last
isotherm is formed very close to the top cold wall.
Fig.5.18 shows the medium is absorbing and scattering. The isotherms at the lower-
left-hand corner are less steep.
Fig.5.19 shows the medium is purely scattering. It takes more time to reach at the
steady state. The isotherms are similar to the isotherms obtained for conduction, fig.(5.1).
Fig.5.20 shows the medium is non-scattering and wall emissivity of εw=0.5. As the wall
emissivity increases, the intensity at the hot wall increases and it further increases the
radiative heat flux and the temperature of the medium inside. Thus the last isotherm is very
closer to the top cold wall.
Fig.5.21 shows the medium is scattering with ω=0.5 and wall emissivity εw=0.5. The
combined effect takes more time to reach at the steady state.
Figure 5.17 Temperature isotherms for N=0.001 and ω=0.
51
Figure 5.18 Temperature isotherms for N=0.001 and ω=0.5.
Figure 5.19 Temperature isotherms for N=0.001 and ω=1.
52
Figure 5.20 Temperature isotherms for N=0.001 and ω=0 and ε=0.5.
Figure 5.21 Temperature isotherms for N=0.001 and ω=0.5 and ε=0.5.
53
5.6 COMPARISION OF ISOTHERMS AT DIFFERENT TIME STEPS:
Temperature isotherms at different time step for N=0.001 and ω=0.5 and ε=0.5 are shown
below.
Figure 5.22 Number of time steps = 20.
Figure 5.23 No of time steps=40.
54
Figure 5.24 No of time steps=80.
5.6.1 Transient Results of temperature: The figure shown below, describes the temperature
variations at different time steps, for N=0.001, ω=0.5 and ε=0.5 at the mid plane. The
physical time required to reach at the steady state can be calculated.
Figure 5.25 Temperature Variation with time at the mid plane
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Dim
ension
less Tem
perature
Dimensionless Coordinate Y
Time Steps=20
Time Steps=40
Time Steps=80
Steady
55
Table 5.1 shows variation in time period required to reach at steady state
Fixed parameters Varying parameters No. of time steps
N=0.001
ε=1
τ=1
ω 0 117
0.5 163
1 1877
ε=1
τ=1
ω=0
N 1 1851
0.1 1606
0.01 624
0.001 117
N=0.001
τ=1
ω=0
εw 0.2 229
0.5 148
0.8 124
1 117
Time required to reach at steady state conditions depends upon ω, ε , and τ, all the radiative
properties and conduction-radiation parameter N. As the value of N decreases the phenomena
will be radiation dominant, thus the period of transient is small. As the emissivity of the wall
increases, the period of transient decreases. As the scattering albedo ω increases the period of
transient increases.
CHAPTER 6
CONCLUSIONS
57
6.1 CONCLUSIONS
Application of the FVM was extended to the solution of the energy equation of a radiation
and Fourier heat conduction problem. The FVM was used to compute the radiative
information. Combined conduction radiation problems with mixed boundary condition
problems were studied. Results of the present work were validated against those available in
the literature. Good agreements were found. Temperature distributions in the medium were
analyzed for different values of the conduction radiation parameter. The results for the
temperature distribution and isothermal contours were illustrated. Additionally fractional
radiative heat flux and total heat flux were also introduced and discussed. The results of the
temperature distribution at different time step were illustrated. Conclusively the finite volume
method is considered to be highly integrable with other finite differenced transport equations.
Furthermore it was found that only a reasonably short computational time was required to
yield quite accurate solutions.
6.2 FUTURE SCOPE
[1] Analysis of non-Fourier conduction-radiation heat transfer in a 2-D and 3-D systems.
[2] Analysis of non-Fourier heat conduction-radiation heat transfer in cylindrical and
spherical system using FVM.
[3] CLAM scheme can be used to solve the radiative transfer equation by FVM.
[4] Convection phenomenon can be added with different boundary condition to the problem.
REFERENCES
59
REFERENCE
1. R. Viskanta and R. J. Grosh, Heat Transfer by Simultanous Conduction and Radiation
in an Absorbing Medium, J. Heat Transfer, vol. 84, no. 1, pp. 63-72, 1962.
2. R. Viskanta, Heat Transfer by Conduction and Radiation in Absorbing and Scattering
Materials, J. Heat Transfer, vol. 87, no. 1, pp. 143-150, 1965.
3. W. W. Yuen and L. W. Wong, Heat Transfer by Conduction and Radiation in a One-
Dimensional Absorbing, Emitting and Anisotropically-Scattering Medium, J. Heat
Transfer, vol. 102, no. 2, pp. 303-307, 1980.
4. P. Talukdar and Subhash C. Mishra, Analysis of Conduction-Radiation Problem in
Absorbing-Emitting Media Using Collapsed Dimension Method, in 34th National
Heat Transfer Conference Pittsburgh, PA, paper no. NHTC2000-12129, 2000.
5. P. Talukdar and Subhash C. Mishra, Analysis of Conduction-Radiation Problem in
Absorbing, Emitting and Anisotropically Scattering Media Using Collapsed
Dimension Method, Int. J. Heat Mass Transfer, submitted.
6. C. C. Lii and M. N. Ozisik, Transient Radiation and Conduction in an Absorbing,
Emitting, Scattering Slab with Re£ective Boundaries, Int. J. Heat Mass Transfer, vol.
15, no. 5, pp. 1175-1179 , 1972.
7. C. Barker and W. H. Sutton, The Transient Radiation and Conduction Heat Transfer
in a Gray Participating Medium with Semi-Transparent Boundaries, in ASME
Radiation Heat Transfer, vol. HTD-49, pp. 25-36, B. F. Armaly and A. F. Emery,
eds., 1985.
8. W. H. Sutton, A Short Time Solution for Coupled Conduction and Radiation in a
Participating Slab Geometry, J. Heat Transfer, vol. 108, no. 2, pp. 465-466, 1986.
9. J. R. Tsai and M. N. Ozisik, Transient, Combined Conduction and Radiation in an
Absorbing, Emitting and Isotropically Scattering Solid Sphere, Journal of Quant.
Spectros. Radiat. Transfer, vol. 38, no. 4, pp. 243-251, 1987.
10. R. Siegel, Finite Difference Solution for Transient Cooling of a Radiating-Conducting
Semitransparent Layer, J. Thermophys. Heat Transfer, vol. 6, no. 1, pp. 77-83, 1992.
11. Chengcai Yao and B. T. F. Chung, Transient Heat Transfer in a Scattering-Radiating-
Conducting Layer, J. Thermophys. Heat Transfer, vol. 13, no. 1, pp. 19-24, 1999.
12. H. Tan, L. Ruan, X. Xia, Q. Yu, and T. W. Tong, Transient Coupled Radiative and
Conductive Heat Transfer in an Absorbing, Emitting and Scattering Medium, Int.
J.Heat Mass Transfer, vol. 42, pp. 2967-2980, 1999.
60
13. H. Tan and M. Lallemand, Transient Radiative-Conductive Heat Transfer in Flat
Glasses Submitted to Temperature, Flux and Mixed Boundary Conditions, Int. J. Heat
Mass Transfer, vol. 32, no. 5, pp. 795-810, 1989.
14. W. W. Yuen and M. Khatami, Transient Radiative Heating of an Absorbing, Emitting
and Scattering Material, J. Thermophys. Heat Transfer, vol. 4, no. 2, pp. 193-198,
1990.
15. J. H. Tsai and J. D. Lin, Transient Combined Conduction and Radiation with
Anisotropic Scattering, J. Thermophys. Heat Transfer, vol. 4, no. 1, pp. 92-97, 1990.
16. T. W. Tong, D. L. McElroy, and D. W. Yarbrough, Conduction and Radiation Heat
Transfer in Porous Thermal Insulations, J. Thermal Insulation, vol. 9, pp. 13-29,
1985.
17. S. W. Baek, T. Y.Kim, and J. S. Lee, Transient Cooling of a Finite Cylindrical
ediumin a Rarefied Cold Environment, Int. J. Heat Mass Transfer, vol. 36, no. 16,
pp. 3949-3956,1993.
18. C. F. Tsai and G. Nixon, Transient Temperature Distribution of a Multilayer
Composite Wall with Effects of Internal Thermal Radiation and Conduction, Numer.
Heat Transfer, vol. 10, no. 1, pp. 95-101, 1986.
19. O. Hahn, F. Raether, M. C. Arduini-Schuster, and J. Fricke, Transient Coupled
Conductive/Radiative Heat Transfer in Absorbing, Emitting and Scattering
Media:Application to Laser-Flash Measurements on Ceramic Materials, Int. J. Heat
Mass Transfer, vol. 40, no. 3, pp. 689-698, 1997.
20. L. A. Diaz and R. Viskanta, Experiments and Analysis on the Melting of a Semi-
Transparent Material by Radiation, Warme-und Stoffubertragung, vol. 20, no.4, pp.
311-321, 1986.
21. M. Nishimura, Y. Bando, M. Kuraishi, and E. W. P. Hahne, Direct Storage of Solar
Thermal Energy Using a Semi-Transparent PCM-Indoor Experiments under Constant
Incidence of Radiation, Int. Chemical Engineering, vol. 29, no. 4, pp. 722-728, 1989.
22. L. K.Matthews, R. Viskanta, and F. P. Incropera, Combined Conduction and
Radiation Heat Transfer in Porous Materials Heated by Intense Solar Radiation, J.
Solar Energy Engineering , vol. 107, no. 1, pp. 29-34, 1985.
23. M. N. Ozisik, Radiative Transfer. Wiley, New York (1973).
24. J.R. Howell, Application of Monte-Carlo to heat transfer problem. In Advances in
Heat Transfer (Edited by T.F Irvine and J.P Hartnett) , Vol. 5, pp. 54, Academic
Press, New York (1968).
61
25. J.J. Noble, The Zone method: explicit matrix relations for total exchange areas, Int. J.
Heat Mass Transfer 18, 261-269(1975).
26. M.M Razzaque, J.R. Howell and D.E Klein, Coupled radiative and conductive heat
transfer in a two-dimensional rectangular enclosure with gray participating media
using finite elements. J. Heat Transfer 106, 613-619(1984).
27. Raithby, G.D., and Chui, E.H., A finite-Volume Method for predicting a Radiant
Heat Transfer in Enclosures with Participating Media,” Journal of Heat Transfer, Vol.
112, No. 2, 1990, pp. 415-423.
28. Chui, E.H., Raithby, G.D., and Huges, P.M.J., “Prediction of Radiative Transfer in
Cylinderical Enclosures with the Finite-Volume Method,” Journal of Thermophysics
and Heat Transfer, Vol. 6, No. 4, 1992, pp. 605-611.
29. Chui, E.H., and Raithby, G.D., “Computation of Radiant Heat Transfer on a
Nonorthogonal Mesh Using the Finite-Volume Method,” Numerical Heat Transfer,
Vol.23, Pt. B, 1993, pp. 269-288.
30. Chai, J.C., Lee, H.S., and Patankar, S.V., “Improved Treatment of Scattering Using
the Discrete Ordinates Method,” Journal of Heat Transfer, Vol. 116, No.1, 1994, pp.
260-263.